Extending user information

I read somewhere (and I can't find it now...) that it's possible to add new attributes to user profiles, and to add new fields to the signup page. I'd like to add a 'referred by' field, so new users can give credit to existing users who referred them to the site.

Can someone point me to the relevant Drupal APIs for this? I'm new to Drupal but not to PHP or SQL, so I'm looking forward to getting into the code.

Thanks!